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Loop condensed phases are scale-invariant quantum liquid phases of matter.These phases include topologically ordered liquid phases such as the toric codeas well as critical liquids such as the Rokhsar-Kivelson point of the quantumdimer model on the square lattice. To investigate the extent to which nonlocalgeometric observables capture a signature of the nonlocal quantum order presentin these phases, we compute geometric properties of such loop condensed statesusing directed loop Monte Carlo calculations. In particular, we investigate theloop condensed nature of ground state of the square lattice quantum dimer modelat the Rokhsar-Kivelson point and compare with other loop condensed states onthe square lattice, including those of the toric code and fully packed loopmodel. The common features of such liquids are a scale invariant distributionof loops and a fractal dimensionality of spanning loops. We find that thefractal dimension of the loop condensate of the square lattice quantum dimermodel at the Rokhsar-Kivelson point is 3/2, which provides quantitativeconfirmation of the effective height model that is commonly used to describethis critical dimer liquid.

Geometric properties of loop condensed phases on the square lattice